Research on nonlinear dynamic characteristics of high-speed gear in two-speed transmission system

The working performance and service life of the two-speed transmission system directly affects the performance and service life of helicopters and other equipment. One of the main tasks of the two-speed transmission system research is to improve its dynamic characteristics. For the two-speed transmission system in high-speed gear, a purely torsional nonlinear dynamic differential equation set considering the number of planetary gears, backlash, and clutch dynamic load is established by using the lumped parameter method, and the equations are dimensionless. Then the dimensionless differential equation set is solved by using the variable step-size fourth-order Runge–Kutta method, and the phase diagram and Poincare diagram of high-speed gear are obtained. By changing the dynamic friction coefficient of the friction clutch and the backlash of the gear pair, the influence of parameter change on the nonlinear dynamic characteristics of the system is analyzed. The results show that, with the increase of excitation frequency, the system has experienced single cycle, quasi-cycle, chaos, and double cycle, then changed from double cycle to chaotic motion, and then changed from chaotic motion to double cycle and single cycle motion in turn, and found the path to chaos. In the low-frequency band, reducing the friction coefficient of the friction clutch can reduce vibration amplitude; In the middle-frequency band, reducing the friction coefficient will make the system tend to unstable vibration. In the high-frequency band, it is a single-cycle movement, which is not affected by friction coefficient.

characteristics of tooth top modification and tooth profile modification of planetary gear train.Ling 11 analyzed the motion and various nonlinear dynamics characteristics of planetary gear system by using global bifurcation diagram, FFT spectrum, Poincare diagram, phase diagram and maximum Lyapunov index.Li 12 established the reliability prediction model of helicopter planetary gear train under partial load.Xiang 13 identified the influence of system motion on the change of backlash by using global bifurcation diagram, maximum Lyapunov index (LLE), FFT spectrum, Poincare diagram, phase diagram and time series.
The research on nonlinear characteristics of planetary gear transmission system mainly focuses on modeling methods, solving methods, stability judgment and other aspects.The models mainly include pure torsional model and bending-torsional coupling model, and the time-varying meshing stiffness, tooth clearance and comprehensive meshing error are usually considered in the modeling.The nonlinear dynamics of planetary gear transmission system can be solved by analytical method and numerical method.
The dynamics of friction clutch is mainly studied by modeling and analyzing the clutch independently or simplifying other mechanical structures.The analysis and solution methods are mainly numerical iteration, concentrated parameters and finite element method.Li 14 analyzed the self-excited vibration characteristics of clutch and discussed the influence of clutch related physical parameters on its performance based on the established 4-DOF nonlinear multi-body dynamics model and Karnopp friction model.Bao 15,16 established the transient thermal analysis model of friction clutch and the motion coupling model in the engagement process, and studied the influence of the groove shape of the friction disc on the transient temperature field in the clutch engagement process and the influence of relevant parameters on the speed and transmission torque of clutch engagement.Wang 17 proposes an improved model for calculating the meshing stiffness of a helical gear system caused by a gear crack, which takes into account the transverse and axial effects of the gear tooth stiffness and the gear foundation stiffness.The results show that the meshing stiffness of the gear is greatly reduced by the existence of cracks.The time domain vibration response of cracked gear is sudden, and the frequency spectrum shows that the more serious the crack, the more abundant the side frequency component and the higher the amplitude.Wang 18 proposed an improved meshing stiffness calculation model for helical gear pairs, which fully considered not only the tooth stiffness of axial gear and the foundation stiffness of axial gear, but also the transverse gear tooth stiffness and foundation stiffness affected by surface roughness under elastohydrodynamic lubrication.The results show that compared with the traditional method, the improved meshing stiffness calculation model can obtain the meshing stiffness under actual lubrication conditions, but the traditional method ignores the axial meshing force and the friction force acting on the gear teeth and gear base.
Compared with the single clutch or planetary gear, the dynamic characteristics of the coupling system composed of friction clutch and planetary gear are more complex.Although this kind of system is widely used, the nonlinear characteristics of friction clutch-planetary gear system are rarely studied.Aleksandar 19 established a simulation model of friction clutch and planetary gear train, and conducted a simulation analysis on the transition process of planetary gear train in the shift process.Michel Bauer 20 established a friction clutch-planetary gear train model suitable for hybrid power and conducted a simulation study on its shifting characteristics.Wang 21 established a planetary gear torsional vibration model considering clutch friction torque and other factors, and studied the influence of clutch friction torque and planetary gear meshing stiffness on system vibration.Chen 22 established a nonlinear dynamics model of two-stage gear transmission system with overrunning clutch, numerically solved the model with Runge-Kutta method, and studied the influence of gear modulus and clutch torsional stiffness on dynamic characteristics of the system.
Through the above analysis, it can be seen that although there have been some studies on the nonlinear characteristics of planetary gear and friction clutch, there are still few studies on the coupling system combined with them, and the influence of the dynamic friction coefficient of friction clutch on the stability of the coupling system is not clear in the research process.Therefore, this paper intends to establish pure torsional nonlinear dynamic differential equations considering the number of planetary wheels, gear clearance and clutch dynamic load by using the lumped-parameter method, and then solve the dimensionless differential equations by using the four-order Runge-Kutta method with variable step size.The influence of dynamic friction coefficient of friction clutch on nonlinear dynamic characteristics of two-speed transmission system is analyzed qualitatively.The research results can provide reference for the optimization design and manufacturing of the subsequent system.

High-gear in-gear dynamics model of a two-speed transmission system
The structure schematic diagram of the two-speed transmission system is shown in Fig. 1.When the friction clutch is released, the planetary frame is connected with the overrunning clutch in the locked state.The power goes through the sun wheel in turn, the middle two planetary wheels, and then the inner gear ring output.At this time, it corresponds to the low gear state.When the friction clutch is engaged, the overrunning clutch is in the overrunning state, and the planetary frame can rotate freely at this time.The power also goes through the sun wheel, the middle two planetary wheels, and the inner gear ring output, which corresponds to the high gear state at this time.In this paper, the nonlinear characteristics of two-speed transmission system are studied for high speed in gear.Its dynamic model is shown in Fig. 2.
In the Fig. 2, R bs is the radius of the solar wheel base circle; R bp1 is the radius of the base circle of the first stage planetary gear.R bp2 is the radius of the base circle of the second planetary gear.R br1 is the radius of the base circle of the inner gear ring; T D is the driving torque; T L is the load moment; J C is the moment of inertia of the planetary frame; θ c is the torsional deformation of the planetary frame; J C1 is the moment of inertia of the friction clutch input end; θ c1 is the torsional deformation of the input end of the friction clutch; J C2 is the rotational inertia of the output end of the friction clutch; θ c2 is the torsional deformation of the output end of the friction clutch; J o is the moment of inertia of the load of the two-speed transmission system; θ o is the torsional deformation of the load shaft of the two-speed transmission system.K sp1 is the time-varying meshing stiffness of the solar wheel and the first stage planetary wheel; C sp1 is the meshing damping of the sun wheel and the first stage planetary wheel; K p1p2 is the time-varying meshing stiffness of the second stage planetary gear and the first stage planetary gear.C p1p2 is the meshing damping between the second stage planetary wheel and the first stage planetary wheel; K p2r1 is the time-varying meshing stiffness between the inner gear ring and the second stage planetary gear.C p2r1 is the meshing damping of the inner gear ring and the second stage planetary gear.

A mathematical model of a two-speed transmission system in high gear
As shown in Fig. 2, K sp1 (t)is the time-varying meshing stiffness of the solar wheel and the first stage planetary gear pair.Its value can be regarded as a rectangular wave, as shown in Eqs.(1)-( 4): (1)  www.nature.com/scientificreports/where, K sp1 is the average value of time-varying meshing stiffness; K sp1r is the amplitude of the RTH harmonic; ϕ sp1r is the phase Angle of the RTH harmonic; ε sp1 is the coincidence degree between the solar gear and the first planetary gear; The first five harmonics can be taken to get a more accurate accuracy, so R = 5.
Assuming that all the gears are unmodified involute spur gears, and ignoring the bending deformation of the input and output shafts, the pure torsional nonlinear mathematical model of the two-speed transmission system as shown in Eq. ( 5) can be derived by using the concentrated parameter method and Newton's law.
Where θ s , θ p1i , θ p2i and θ r1 are respectively the torsional vibration displacements of the sun wheel, the i planetary wheel of the first stage, the i planetary wheel of the second stage and the inner gear ring (i = 1, 2, N).Differentiation concerning time; Input torque T s (t) is the fluctuation value, which can be expressed as T s (t) = T sm + T saT (t), where T sm is the mean of the torque, T saT (t) is the instantaneous fluctuation value, and can be expressed as T saT (t) = T saT sin(ω aT t + ϕ aT ); e sp1i (t) is the static transmission error between the sun wheel and the first stage i planetary wheel, and is related to the manufacture and assembly of gears.It can be regarded as e sp1i (t) = êsin(ω e t + ϕ e ).T c1 is the friction torque of the inner gear ring and the planetary frame, which is related to the pressure applied by the friction clutch, the number of friction plates and the friction coefficient of the friction plates, as shown in Eq. (6), where R o is the radius of the outer circle of the friction plate, and R i is the radius of the inner circle of the friction plate.T L (t) is the load torque applied on the output shaft of the driveline.
where x s , x p1i , x p2i , x r1 , x c1 , x c2 , x o are the equivalent linear displacements of the solar wheel, the planetary wheel 1, the planetary wheel 2, the inner gear ring, the input end of the friction clutch, the output end of the friction clutch, and the output shaft, respectively.By defining new variables The dimensionless time scale τ = taa n and the displacement scale b c are defined.

Parameter study
The basic parameters of the two-speed transmission system are shown in Table 1.The Runge-Kutta method with fourth-order variable step size is used to solve the equation set 7. The initial values of all displacements are 0, and the initial values of all velocities are 0.01, and the solution interval is [0, 600t].

The influence of excitation frequency
Figure 3 shows the dimensionless torsional vibration bifurcation diagram of the inner gear ring drawn with the excitation frequency of ωh as the control variable in the high speed mode of the two-speed transmission system.It can be seen from the Fig. 3 that there is an obvious jump phenomenon at ωh = 0.65 and at ωh = 0.87 of the inner gear ring.When 0.65 ≤ ωh ≤ 0.87, the inner gear ring moves in a single periodic motion, and the vibration increases with the increase of the excitation frequency.When the excitation frequency is in the low frequency band (ωh ≤ 0.65 and 0.87 ≤ ωh ≤ 1.02) and the high frequency band (ωh ≥ 2.0), the torsional vibration is a relatively stable single periodic motion, which is not affected by the excitation frequency.When the excitation frequency is in the range of 1.04 ≤ ωh ≤ 1.25, 1.41 ≤ ωh ≤ 1.87, the inner gear ring enters the chaotic region.
When the excitation frequency is within 1.27 ≤ ωh ≤ 1.4, the motion of the system is a double cycle motion.In the range of 1.88 ≤ ωh ≤ 1.98, the movement of the system changes from double period to single period.Figure 4 shows the time domain diagram, phase diagram and Poincare cross section diagram of the system at the excitation frequency point ωh = 0.7633, 1.6196, 1.9332 and 2.2588, which respectively correspond to quasi-periodic, chaotic, double cycle and haploid periodic motions.

Influence law of dynamic friction coefficient of friction clutch
Figure 5 shows the influence of dynamic friction coefficient of friction clutch on bifurcation characteristics of internal gear ring.As can be seen from the figure, with the increase of the dynamic friction coefficient, the oscillation value of the system in the unstable region can be effectively suppressed.From Fig. 5a-c, it can be seen that the system experienced quasi-periodic motion and haploperiodic motion in the whole excitation frequency range.

Conclusion
The dynamic differential equation of pure torsional nonlinear dynamic load is established by lumped parameter method, and the parameters of planetary wheel number, backlash and clutch dynamic load are considered in the modeling process, and the equations are dimensionless.Then the fourth order Runge-Kutta method with variable step size is used to solve the dimensionless differential equation, and the phase diagram and Poincare diagram in the high-speed gear file are obtained.By changing the friction coefficient of the friction clutch, the influence of parameter change on the nonlinear dynamic characteristics of the system is analyzed, and the following conclusions are obtained: (1) With the increase of excitation frequency, the system went through a single period, a quasi period, chaos, a double period, and then the second period turned into chaotic motion, and then the second period turned into a double period and a single period motion, and the path to the chaos was found out.To make the system have good dynamic characteristics, the excitation frequency of the system should be guaranteed to be small or large.(2) In the low-frequency band (ωh ≤ 0.9), the friction coefficient of the friction clutch can be reduced to reduce the vibration amplitude; In the middle-frequency band (0.9 ≤ ωh ≤ 1.9), reducing the friction coefficient will make the system tend to be unstable vibration.In the high-frequency band (1.9 ≤ ωh ≤ 2.5), it is a single-times periodic motion, which is not affected by the friction coefficient.In order to make the system have good stability and reduce the vibration of the system during operation, when the system is in the low-frequency stage, it is more appropriate to select a smaller value for the friction coefficient of the friction clutch; When the system is in the intermediate frequency stage, it is more appropriate to take a larger value for the friction coefficient of the friction clutch; When the system is in the high-frequency phase, the friction coefficient can be optionally selected because the coefficient of friction has little effect on the stability of the system.

Limitations and deficiencies
(1) This paper adopts the lumped parameter method in modeling and the Runge-Kutta method in solving differential equations, which is a commonly used modeling and solving method for system dynamics.However, the lumped parameter method is a simplified modeling method, which usually regards the shaft  as a point mass or a rigid connection, and has certain limitations in capturing the deflection and deformation of the shaft.(2) In future studies, under the condition that the amount of calculation is appropriate, if the accuracy is required to be high, the potential energy method or Timoshenko theory can be considered for modeling, which will make the results more accurate.

Figure 2 .
Figure 2. High-speed gear dynamics model of two-speed transmission system.

Figure 3 .
Figure 3. Global bifurcation diagram of torsional vibration of internal gear ring varying with excitation frequency ωh.

Table 1 .
Two-speed transmission system parameters.